Direct Variation
Algebra I Mastery
Cate Condon
The store you go into in the mall sells t-shirts. You
are looking around and you see that 3 t-shirts cost
$15 total. Given this information how much would
5 t-shirts cost?
O We can set up a proportion:
3
5
=
15
c
O Cross multiply:
3c = 5*15= 75
3c = 75
c = 25
O 3c = 75
O We know that c is the variable. What does
the c stand for or represent?
O What is the 3 in front of the c called?
O 3c = 75
O We know that 3 is called the coefficient of
the variable.
O Another name for this is the constant of
variation.
O Why do you think that a number in front of
variable is called a constant of variation?
O 10 = 2x  2x = 10
O For this equation, what is the variable and
what is the constant of variation?
O 3x = 1/6
O For this equation, what is the variable and
what is the constant of variation?
O 10 = 2x
O We can write this equation substituting
variables: y = kx where k is the constant of
variation and x and y are variables
O We call this equation, y = kx, direct variation
Direct Variation
O A direct variation is a function in the form
y = kx where k does not equal 0.
O An equation is a direct variation if the
equation can be written in the form y = kx.
Is the Equation a Direct Variation?
If it is, find the constant of variation.
O -6x + 2y = 0
Want to see if the
equation can be written like y=kx
O Solve for y:
2y = 6x
y = 3x
O Yes it is a direct variation.
O The constant of variation is 3
Is the Equation a Direct Variation?
If it is, find the constant of variation
O 7y = 2x
O Solve for y:
y = (2/7) x
O Yes it is a direct variation.
O The constant of variation is (2/7)
Try These
O 0 = 10 + 3x
O 8x = 4y
O 6 + 9x = 2y
Direct Variation
O We say that y varies directly with x when we
have the equation y = kx.
O So if I say that p varies directly with z, what
would our equation be? (keep k as our
constant)
Real World Example
O Your distance from lightning varies directly
with the time it takes you to hear thunder. If
you hear thunder 10 seconds after you see
the lightning, you are about 2 miles from the
lightning. Write an equation for the
relationship between time and distance.
Real World Example
O Your distance from lightning varies directly
with the time it takes you to hear thunder:
O y = distance from lightening
O x = time it takes you to hear thunder
Real World Example
O y = kx
O We know that y is 2 miles from the lightning (y =
distance from lightening
O We know that x is 10 seconds (x = time it takes
you to hear thunder)
O 2 = k(10)  solve for k
O k = (2/10)=(1/5)
O Our equation: y = (1/5)x
Real World Example
y
x
k
Case 1
2
10
1/5
Case 2
3
15
1/5
Case 3
4
20
1/5
Case 4
5
25
1/5
DV
equation
y=
(1/5)*10
y=
(1/5)*15
y=
(1/5)*20
y=
(1/5)*25
Real World Example
30
25
20
y
x
k
15
10
5
0
0
1
2
3
4
5
Real World Example
Real World Example
Try This
O A recipe for a dozen corn muffins calls for 1
cup of flower. The number of muffins varies
directly with the amount of flour you use.
Write a direct variation for the relationship
between the number of cups of flour and the
number of muffins
Real World Example
O y = kx
O y = the number of muffins
O x = amount of flour used
O 12 = k(1)  12 = k
O So our equation is y = 12x
Closing
O p varies directly with z. If p = 210 when z =
200, then write the formula for the relation
between p and z.
O Work by yourself on the notecard on your
desk for the last 5 minutes.